Crossing symmetry leads to the set of sum rules which were dubbed null constraints in Ref. [1]}. Effectively, they express the low-spin partial-wave data in terms of the higher-spin data. Consider for example the relation \(a_{1,1}=0\) , listed in Eq. (REF ). Using the formulas above we can write it as follows
\(\int _{m_{\rm gap}^2}^\infty {d m^2 \over (m^2)^{2}} {\rho _{4}^{+-}(m^2) \over m^8} = \int _{m_{\rm gap}^2}^\infty {d m^2 \over (m^2)^{2}} \left( \sum _{J=2}^\infty {1+(-1)^J \over 2} {\rho _{J}^{++}(m^2) \over m^8} {\cal J}^2 + \sum _{J=5}^\infty {\rho _{J}^{+-}(m^2) \over m^8} ({\cal J}^2 - 21) \right) ,\) 
